This talk is based on a joint work with Hironori Kumura (Shizuoka University, Japan). The classical Hardy inequality for the Laplacian $\Delta = \mathrm{div}\nabla$ on $\mathbb{R}^n$ shows the borderline-behavior of a potential $V$ for the following question : whether the Schrödinger operator $-\Delta + V$ has a finite or infinite number of the discrete spectrum. In this talk, we will show a sharp generalization of this inequality on $\mathbb{R}^n$ to a relative version of that on large classes of complete noncompact manifolds. Replacing $\mathbb{R}^n$ by some specific classes of complete noncompact manifolds, including hyperbolic spaces, we also show some sharp criteria for the above-type question.